I can that this is showing . But I can't see how this equality is equal to or

May 21st 2012, 10:52 AM

alyosha2

Re: Proving inequality through induction

I understand now the equality sign only applies to the expression on the side of the inequality sign on which it is placed, rather than equating the whole inequality expression.

Given the explanation posted in the thread I'm still confused about why the working I have posted gives the inequality to larger than 4k - 4 rather than 4(K+1).

Is this something to do with the shifting of the inequality sign?

Why have they chosen to express the ineqaulity sign as they have?

May 22nd 2012, 06:33 AM

alyosha2

Re: Proving inequality through induction

After many hours of intense thought I came to a solution and thought I should post it so as to not to leave the thread without clousre in case someone has a similar problem. Also I hope it will help crystallise the idea for me and perhaps I've even got it wrong so others could refine it.

First a word about precedence and equality signs. It's usual to think of the equality sign as having the lowest possible precedence.This however is not the case when considering inequality signs which appear to have an even lower precedence than equality signs. So a = b > c = d is (a = b) > (c = d) or just b > c and not a = (b > c) = d.

This means that if is less than then subtracting the smaller from the larger would leave a difference that is larger than 0.

Now to show that the equality will hold we will have to show that if the difference left by then the difference left by .

How can we do this?

Well we use to express the difference of the proposition with k+1 and note that this can be expressed in terms of k as .

Now if we assume that http://latex.codecogs.com/png.latex?2%5Ek%20%3E%204k we can use this inequality to find an expression that the difference of the proposition with (K+1) will be larger than and if this expression is larger than 0 we know the proposition will hold.

We do this by manipulating the left side of the inequality to become and then do the same to the other the other side of the inequality to maintain equality with original.

So

The left hand side of this equality simplifes to and then to and this expression will equate to greater than 0 for k > 1. So we know that the proposition will hold for some value k > 1.

This is my understanding and it seems satisfactory as far as I can tell. Please let me know if I have made any wrong assumptions or have completely misunderstood.

Thanks.

May 22nd 2012, 10:13 AM

Deveno

Re: Proving inequality through induction

you have some errors in your proof, but the idea is almost right.

we want to prove that if n > 4, then 2n > 4n.

we want to prove this for all natural numbers greater than 4. the "way" we do this is prove it for n = 5, and then show: